They did not lie to you. You messed up. Sorry to be upfront. You took the situation lightly and did not check what the restrictions were and what you needed to present in order to be admitted. You even messed up the PCR timing (72hs before arrival). LATAM also messed up - -they should not have let you board that plane.Why would migraciónes lie to me? Why would they care?
Am I supposed to leave after 3 months like on a normal tourist visa? Is there any bar to entering and re-entering after a couple of weeks like I used to do?
When you arrive in ARG, apply for permanent residency in Radex as you have a ARG child. If I remember correctly you already started the process in Radex? If not do it when you arrive. Migraciones is currently very quick in giving out Precarias (generally within 48hs of submission of ALL the documents). With the precaria you don't have to leave every 90 days.
As Bajo pointed out, you are also eligible for ARG citizenship. You can start the process too if you want. But that process goes thru the courts/Tribunales not Migraciones. It will take 18-24 months. Good luck.